Exercise 4

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You can start working on your copy of Exercise 4 by accepting the GitHub Classroom assignment.

Exercise 4 is due by the end of the day Tuesday in week 5 of the course.

You can also take a look at the open course copy of Exercise 4 in the course GitHub repository (does not require logging in). Note that you should not try to make changes to this copy of the exercise, but rather only to the copy available via GitHub Classroom.

Hints for Exercise 4

Problem 1: Calculating dh/dt (see also hint below)

For the calculate_dhdt function, you should ensure the value you return is negative. The stream-power itself can be positive, but returning a negative value ensures the stream-power equation acts to lower the channel elevations.

\[\begin{equation} \Large \frac{dh}{dt} = -K A^{m} S^{n} \end{equation}\]

Problem 1: Calculating the channel slope

In order to be able to calculate stream-power erosion from a river channel profile we first need to estimate both the upstream drainage area and calculate the topographic slope. You have hopefully already calculated the drainage area using your drainage_area function, and now we can turn our focus to the slope. Calculating the slope is not necessarily a difficult thing with NumPy, but you do need to be careful how you do it.

The first thing to recognize is that the slope at any given point will simply be the difference in elevation between that point and a neighboring point divided by the horizontal distance between them. In other words, slope is the change in elevation divided by the change in horizontal distance. You could calculate this by hand, but NumPy has some useful functions for this, including np.diff. Below, you can see how to use np.diff to calculate the channel slope.

slope = np.diff(topography)/dx           # Calculate profile slopes; diff() calculates elevation difference between points
slope = abs(slope)                       # Take the absolute value of the slope to avoid direction issues
slope = np.append(slope, slope[-1])      # Append one extra slope value to be same size as the area array

Above, we use np.diff to calculate the slope, but since it calculates the difference between the elevation of points, it returns one less value than in topography. To account for this, we add one extra value back to the slope variable.

Problem 2: Units

Just as a reminder, the values for dhdt are m/a and the plot should have units of mm/a. We also want the erosion rate to plot as a positive value, rather than the negative values that dhdt has in the code. You can compensate for these differences when setting the y data to plot.